Optimal. Leaf size=38 \[ \frac {B (b+c x)^5}{5 c^2}-\frac {(b+c x)^4 (b B-A c)}{4 c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \[ \frac {B (b+c x)^5}{5 c^2}-\frac {(b+c x)^4 (b B-A c)}{4 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^3} \, dx &=\int \left (\frac {(-b B+A c) (b+c x)^3}{c}+\frac {B (b+c x)^4}{c}\right ) \, dx\\ &=-\frac {(b B-A c) (b+c x)^4}{4 c^2}+\frac {B (b+c x)^5}{5 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 67, normalized size = 1.76 \[ A b^3 x+\frac {1}{2} b^2 x^2 (3 A c+b B)+\frac {1}{4} c^2 x^4 (A c+3 b B)+b c x^3 (A c+b B)+\frac {1}{5} B c^3 x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.81, size = 69, normalized size = 1.82 \[ \frac {1}{5} \, B c^{3} x^{5} + A b^{3} x + \frac {1}{4} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + {\left (B b^{2} c + A b c^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 72, normalized size = 1.89 \[ \frac {1}{5} \, B c^{3} x^{5} + \frac {3}{4} \, B b c^{2} x^{4} + \frac {1}{4} \, A c^{3} x^{4} + B b^{2} c x^{3} + A b c^{2} x^{3} + \frac {1}{2} \, B b^{3} x^{2} + \frac {3}{2} \, A b^{2} c x^{2} + A b^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 73, normalized size = 1.92 \[ \frac {B \,c^{3} x^{5}}{5}+A \,b^{3} x +\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{4}}{4}+\frac {\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{3}}{3}+\frac {\left (3 A \,b^{2} c +b^{3} B \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.90, size = 69, normalized size = 1.82 \[ \frac {1}{5} \, B c^{3} x^{5} + A b^{3} x + \frac {1}{4} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + {\left (B b^{2} c + A b c^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 65, normalized size = 1.71 \[ x^2\,\left (\frac {B\,b^3}{2}+\frac {3\,A\,c\,b^2}{2}\right )+x^4\,\left (\frac {A\,c^3}{4}+\frac {3\,B\,b\,c^2}{4}\right )+\frac {B\,c^3\,x^5}{5}+A\,b^3\,x+b\,c\,x^3\,\left (A\,c+B\,b\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.08, size = 73, normalized size = 1.92 \[ A b^{3} x + \frac {B c^{3} x^{5}}{5} + x^{4} \left (\frac {A c^{3}}{4} + \frac {3 B b c^{2}}{4}\right ) + x^{3} \left (A b c^{2} + B b^{2} c\right ) + x^{2} \left (\frac {3 A b^{2} c}{2} + \frac {B b^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________